How do I create a material that emits light based on how much light hits it?
This is how every material works:


Light hits material
Light might get absorbed by material (some wavelengs) and turned into heat 
The rest of the light is reflected into different dirrections (it is diffused, reflected, refracted, bounces inside volume, etc.)


So the answer is easy, if you need to double the light that comes off the surface, you just add 2 shaders together (with add shader). This will break the energy conservation, but that is what you want.


If you need to emit the light in all directions, add white diffuse shader to your material.
If you need to emit the light directionally (as if bounced off of mirror), add white sharp glossy to whatever material you have. You can alter the sharpness of the light with rougness and also direction by manipulating the surface normals.


It is very possible that you can get some strange results.